> Note: definable is short for "definable by parameter free finite> formula"

What do you believe to gain by parameters in definitions?Do you want to define the real numbers by using real numbers asparameters?

Otherwise you will not arrive at uncountably many definitions, you mayuse all natural numbers and all rational numbers. Or has some greatleading matheologian who bolsters his ego by calling himself a logicantold you that by parametric definitions the real numbers all can bedefined?

Be informed, that is not the case. If there are uncountable many realnumbers, then they are not real in that nearly all are completelyundefined. Therefore they cannot be used in Cantor lists and cannotspring off as diagonals. They would be completely unrelated toCantor's "proof" and to mathematics.